The tag convex-geometry currently (2019/02/23) has the usage guidance: A branch of geometry dealing with convex sets and functions. Polytopes, convex bodies, discrete geometry, linear programming, antimatroids, ... Also, convexity and convex-analysis have no usage guidance at this date.

Vaguely:

How could this be improved?

Adding tag guidance would require to have a more precise idea how to distribute the roles of these tags. But I see no coherent way to do so.

I see two essential meanings for "convexity": convexity of functions (closer to convex analysis) and convexity of bodies (closer to convex geometry, which also encompasses convexity in metric spaces). The distinction is clear in a number of cases, but this is also well-intertwined since convex functions are usually defined on convex subsets, while convex sets can usually be defined as sub-level sets of convex functions.

At this moment, a quick look at the actual use of convexity seems to be a slight majority of questions on convex functions, an a large minority of convex geometry questions, and it's regularly used in combination with one of the other two tags (40%? I made no serious count).

At this point I see among the various options:

Keep these 3 tags. I guess this considers that convexity is the union of convex-analysis and convex-geometry. Improve usage guidance accordingly (how?) to avoid a too random use. Clarify when should convexity be used: when both others are applicable? when at least one is applicable?

"convexityis perplexing since it's almost a meta-tag. My preference would be to have these questions remapped to convex-analysis, convex-geometry, and similar. However, this seems very implausible at the moment."

The closest option above is the technically easy (2), solving the issue for future questions. If one wants to solve the issue for past questions, if one wants to completely delete convexity, one could imagine (if technically doable): remove convexity from every question with at least one of the other 2 tags; for those tagged convexity and none of the other 2, remove convexity and add the other two (this will block for a few remaining ones for which there are already 5 tags but one can imagine these few can be dealt manually). But this can sound arbitrary, so just deprecating sounds enough, while users can then progressively manually delete past occurrences of deprecated convexity.

$\begingroup$I suppose you saw a comment by François G. Dorais♦ regarding this issue in the "Help improve tagging!" thread. I hope he will join into this discussion - he is the moderator who most frequently deals with tag-related issues.$\endgroup$
– Martin SleziakFeb 24 at 19:38

2

$\begingroup$It's pretty clear that convexity should disappear and needs to be deprecated; this will happen soon regardless. That said, convexity is a very complicated case and I am looking forward to what comes out of this discussion. YCor, if you don't mind, I can add a few comments to your post (clearly labeled as from me) to help guide the discussion.$\endgroup$
– François G. Dorais♦Feb 25 at 2:38

$\begingroup$@FrançoisG.Dorais When you mention deprecating (convexity) in your comment, do you mean simply adding into the tag-excerpt that it is deprecated (similarly as for some other tags in the past), or also blacklisting the tag - which would prevent it from being used in the new questions? See also: The existing deprecated tags should be blacklisted.$\endgroup$
– Martin SleziakFeb 25 at 5:27

$\begingroup$@MartinSleziak The tag-excerpt option only. Blacklisting is mostly useful for repeat offenders: deprecated tags that get created over and over again. That problem is very rare on MO.$\endgroup$
– François G. Dorais♦Feb 25 at 6:04